Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schr\"odinger equations with repulsive nonlinearities

TL;DR

This paper introduces a new iterative method and preconditioner for efficiently solving linear systems arising from space fractional coupled nonlinear Schr"odinger equations, demonstrating improved computational performance and spectral properties.

Contribution

The paper proposes a novel diagonal and normal with Toeplitz-block splitting iteration method and circulant-block preconditioner, with theoretical convergence proof and eigenvalue bounds for fractional Laplacian matrices.

Findings

Unconditional convergence of the new iteration method.

Significant improvement in computational efficiency with the preconditioner.

Linear dependence of GMRES iteration count on space mesh size, decreasing with fractional order.

Abstract

By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schr\"odinger equations leads to a sequence of linear systems with complex symmetric and Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and normal with Toeplitz-block splitting iteration method to solve the above linear systems. The new iteration method is proved to converge unconditionally, and the optimal iteration parameter is deducted. Naturally, this new iteration method leads to a diagonal and normal with circulant-block preconditioner which can be executed efficiently by fast algorithms. In theory, we provide sharp bounds for the eigenvalues of the discrete fractional Laplacian and its circulant approximation, and further analysis indicates that the spectral distribution of the preconditioned system matrix is tight. Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods. Moreover, the behavior of the corresponding preconditioned GMRES method exhibits a linear dependence on the space mesh size, which weakens as the fractional order parameter decreases.